► We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves…
(more)

▼ We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to
the setting of Hilbert modular forms. Our work involves three parts. In the rst part, we construct
Eisenstein series adelically and compute their constant terms by computing local integrals. In the
second part, we prove a control theorem for one-variable ordinary Λ-adic modular forms following
Hida's work on the space of multivariable ordinary Λ-adic cusp forms. In part three, we compute
congruence modules related to Hilbert Eisenstein series through an analog of Ohta's methods.
Advisors/Committee Members: Sharifi, Romyar (advisor), Cais, Bryden (committeemember), Xue, Hang (committeemember), Levin, Brandon (committeemember).

► In this paper we report the beginnings of the computations and tabulations of the generators of \PSL2(\OK), where \OK is the maximal order of a…
(more)

▼ In this paper we report the beginnings of the
computations and tabulations of the generators of \PSL2(\OK),
where \OK is the maximal order of a real field of degree
n=[K:\QQ]. We discuss methods of obtaining generators in order to
calculate the values of invariants of the congruence subgroups.
[This text field cannot display some of the mathematical formatting
used in this abstract. To see the correctly formatted abstract,
please click on the PDF below.]; Generators, Hilbert, Modular,
Quadratic fields
Advisors/Committee Members: Sebastian Pauli (advisor).

Everhart LM. On generators of Hilbert modular groups of totally real
number fields. [Masters Thesis]. University of North Carolina – Greensboro; 2016. Available from: http://libres.uncg.edu/ir/listing.aspx?styp=ti&id=21341

► In this paper we report the beginnings of the computations and tabulations of the generators of \PSL2(\OK), where \OK is the maximal order of a…
(more)

▼ In this paper we report the beginnings of the computations and tabulations of the generators of \PSL2(\OK), where \OK is the maximal order of a real field of degree n=[K:\QQ]. We discuss methods of obtaining generators in order to calculate the values of invariants of the congruence subgroups. [This text field cannot display some of the mathematical formatting used in this abstract. To see the correctly formatted abstract, please click on the PDF below.]

► Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full Hilbert C*-modules over A and B,…
(more)

▼ Let A and B be real or complex C*-algebras. Let V and W be real or complex (right) full
Hilbert C*-modules over A and B, respectively. Let T be a linear bijective map from V onto
W. We show the following four statements are equivalent.
(a) T is a unitary operator, i.e., there is a ∗-isomorphism Î± : A â B such that
<Tx,Ty> = Î±(<x,y>), ∀ x,y∈ V ;
(b) T preserves TRO products, i.e., T(x<y,z>) =Tx<Ty,Tz>, ∀ x,y,z in V ;
(c) T is a 2-isometry;
(d) T is a complete isometry.
Moreover, if A and B are commutative, the four statements are also equivalent to
(e) T is a isometry.
On the other hand, if V and W are complex Hilbert C*-modules over complex C*-algebras,
then T is unitary if and only if it is a module map, i.e.,
T(xa) = (Tx)Î±(a), ∀ x ∈ V,a ∈ A.
Advisors/Committee Members: Man-Duen Choi (chair), Mau-Hsiang Shih (chair), Hwa-Long Gau (chair), Mao-Ting Chien (chair), Pei-Yuan Wu (chair), Ngai-Ching Wong (committee member), Chi-Kwong Li (chair).

► Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and…
(more)

▼ Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels.
Advisors/Committee Members: Ball, Joseph A. (committeechair), Rossi, John F. (committee member), Floyd, William J. (committee member), Elgart, Alexander (committee member).

► The coordinate rings of the classical determinantal varieties are each isomorphic to a classical invariant ring by Weyl's fundamental theorems of invariant theory. Since these…
(more)

▼ The coordinate rings of the classical determinantal varieties are each isomorphic to a classical invariant ring by Weyl's fundamental theorems of invariant theory. Since these rings are Cohen-Macaulay, their Hilbert series are rational functions whose numerator polynomials have nonnegative integer coefficients. In the case of general determinantal varieties, as well as in the case of symmetric determinantal varieties, these numerator polynomials were shown to be equal to the Hilbert series of certain finite-dimensional highest weight modules and were given an explicit combinatorial description. The current work extends these results to the alternating determinantal varieties. The proof of these results, in all three cases, relies on the fact that the coordinate rings of the determinantal varieties carry the structure of a Wallach representation. The Hilbert series of the Wallach representation is a rational function whose numerator polynomial is given by the Hilbert series of a finite-dimensional highest weight module, and the Hilbert series of the determinantal variety is equal to the Hilbert series of the Wallach representation. T. J. Enright and J. F. Willenbring introduced the more general class of quasi-dominant weights and showed that if L is a unitarizable highest weight module with quasi-dominant highest weight, then the Hilbert series of L is of the form H_L(t) = R * (H_E(t)) / ((1-t)^D), where R is a rational number, E is a finite-dimensional highest weight module, and D is the Gelfand-Kirillov dimension of L. The set of quasi-dominant weights has an interesting characterization in terms of parabolic category O and Kostant's minimal length coset representatives. We give a new characterization in terms of associated varieties and show that the subset of quasi-dominant weights whose highest weight modules occur in the setting of Howe dual pairs has a nice description in terms of the highest weights of the "Howe dual'' representations. Finally, we give some new results on the number of quasi-dominant reduction points.
Advisors/Committee Members: Hunziker, Markus, 1968- (advisor).

► The subject of this thesis lies at the junction of mainly three topics: construction of large families of Arithmetically Cohen-Macaulay indecomposable vector bundles on a…
(more)

▼ The subject of this thesis lies at the junction of mainly three topics: construction
of large families of Arithmetically Cohen-Macaulay indecomposable vector bundles
on a given projective variety X, the shape (i.e, the graded Betti numbers) of
the minimal free resolution of a general set of points onX and the (ir)reducibility
of the Hilbert scheme Hilbs(X) of zero-dimensional subschemes Z (belongs) X of length
s.
(Fore more details see the Full Summary enclosed as a complementary file)
Advisors/Committee Members: [email protected] (authoremail), false (authoremailshow), Arrondo, Enrique (director), Miró-Roig, Rosa M. (Rosa Maria) (director), Miró-Roig, Rosa M. (Rosa Maria) (tutor), true (authorsendemail).

▼ In his paper “Generalized Fixed Point Algebras and Square-Integrable Group Actions,” Ralf Meyer showed how to construct generalized fixed-point algebras for C*-dynamical systems via their square-integrable representations on Hilbert C*-modules. His method extends Marc Rieffel’s construction of generalized fixed-point algebras from proper group actions. This dissertation seeks to generalize Meyer’s work to construct generalized fixed-point algebras for twisted C*-dynamical systems. To accomplish this, we must introduce some brand-new concepts, the foremost being that of a twisted Hilbert C*-module. A twisted Hilbert C*-module is basically a Hilbert C*-module equipped with a twisted group action that is compatible with the module’s right C*-algebra action and its C*-algebra-valued inner product. Twisted Hilbert C*-modules form a category, where morphisms are twisted-equivariant adjointable operators, and we will establish that Meyer’s bra-ket operators are morphisms between certain objects in this category. A by-product of our work is a twisted-equivariant version of Kasparov’s Stabilization Theorem, which states that every countably generated twisted Hilbert C*-module is isomorphic to an invariant orthogonal summand of the countable direct sum of a standard one if and only if the module is square-integrable. Given a twisted C*-dynamical system, we provide a definition of a relatively continuous subspace of a twisted Hilbert C*-module (inspired by Ruy Exel) and then prescribe a new method of constructing generalized fixed-point algebras that are Morita-Rieffel equivalent to an ideal of the corresponding reduced twisted crossed product. Our construction generalizes that of Meyer and, by extension, that of Rieffel. Our main result is the description of a classifying category for the class of all Hilbertmodules over a reduced twisted crossed product. This implies that every Hilbert module over a d -dimensional non-commutative torus can be constructed from a Hilbert space endowed with a twisted ℤd -action and a relatively continuous subspace.
Advisors/Committee Members: Sheu, Albert J-L (advisor), Lerner, David E (cmtemember), Martin, Jeremy L (cmtemember), Torres, Rodolfo H (cmtemember), Ralston, John P (cmtemember).

We give a thorough account of the various equivalent notions for \sheaf" on a locale, namely the separated and complete presheaves, the local home- omorphisms,…
(more)

▼

We give a thorough account of the various equivalent notions for \sheaf"
on a locale, namely the separated and complete presheaves, the local home-
omorphisms, and the local sets, and to provide a new approach based on
quantale modules whereby we see that sheaves can be identi¯ed with certain
Hilbertmodules in the sense of Paseka. This formulation provides us with
an interesting category that has immediate meaningful relations to those of
sheaves, local homeomorphisms and local sets.
The concept of B-set (local set over the locale B) present in [3] is seen
as a simetric idempotent matrix with entries on B, and a map of B-sets as
de¯ned in [8] is shown to be also a matrix satisfying some conditions. This
gives us useful tools that permit the algebraic manipulation of B-sets.
The main result is to show that the existing notions of \sheaf" on a locale
B are also equivalent to a new concept what we call a Hilbert module with
an Hilbert base. These modules are the projective modules since they are
the image of a free module by a idempotent automorphism
On the ¯rst chapter, we recall some well known results about partially ordered sets and lattices.
On chapter two we introduce the category of Sup-lattices, and the cate-
gory of locales, Loc. We describe the adjunction between this category and
the category Top of topological spaces whose restriction to spacial locales give us a duality between this category and the category of sober spaces. We
¯nish this chapter with the de¯nitions of module over a quantale and Hilbert
Module.
Chapter three concerns with various equivalent notions namely: sheaves
of sets, local homeomorphisms and local sets (projection matrices with entries
on a locale). We ¯nish giving a direct algebraic proof that each local set is
isomorphic to a complete local set, whose rows correspond to the singletons.
On chapter four we de¯ne B-locale, study open maps and local homeo-
morphims.
The main new result is on the ¯fth chapter where we de¯ne the Hilbertmodules and Hilbertmodules with an Hilbert and show this latter concept
is equivalent to the previous notions of sheaf over a locale.

► The thesis consists of two parts of work. In the first part, we study the geometry of the Hilbert schemes of points on singular curves…
(more)

▼ The thesis consists of two parts of work. In the first part, we study the geometry of the Hilbert schemes of points on singular curves and surfaces. In particular, the Hilbert scheme is irreducible for a surface with only cyclic quotient rational double point singularities. In the second part, we construct new examples of Kodaira non-vanishing on surfaces in positive characteristics and deduce further pathologies on their linear systems.
Advisors/Committee Members: Ein, Lawrence (advisor), Tucker, Kevin (committee member), Coskun, Izzet (committee member), Skalit, Chris (committee member), Mustata, Mircea (committee member).

► Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc.…
(more)

▼ Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*-modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*-modules leaves a frame or a non-frame set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*-modules may possess infinitely many alternative duals due to the existence of zero-divisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert C^*-modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*-algebra is a commutative W*-algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also prove the existence and uniqueness of the best Parseval multi-frame approximations for multi-frame generators of unitary groups on Hilbert C*-modules when the underlying C*-algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*-module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*-modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*-module. For the perturbation of frames and Riesz bases in Hilbert C*-modules we prove that the Casazza-Christensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*-modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a…
Advisors/Committee Members: Han, Deguang.

► A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is completely determined by a C*-correspondence, which consists of a right Hilbert A-…
(more)

▼ A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is
completely determined by a C*-correspondence, which consists of a right Hilbert A-
module, E, and a *-homomorphism from the C*-algebra A into L(E), the adjointable
operators on E. Some familiar examples of C*-algebras which can be recognized as
Cuntz-Pimsner algebras include the Cuntz algebras, Cuntz-Krieger algebras, and
crossed products of a C*-algebra by an action of the integers by automorphisms.
In this dissertation, we construct a Cuntz-Pimsner Algebra associated to a dynam-
ical system of a substitution tiling, which provides an alternate construction to the
groupoid approach found in [3], and has the advantage of yielding a method for com-
puting the K-Theory.
Advisors/Committee Members: Putnam, Ian Fraser (supervisor).

…functions. Chapter 2 is devoted to providing some necessary
background on Hilbertmodules as is… …require the use of a computer to complete.
3
Chapter 2
Background
2.1
HilbertModules
A… …module theory. Since Hilbertmodules generalize Hilbert
spaces, we can often use our intuition… …gained from Hilbert spaces to guide us in our
understanding of Hilbertmodules, but we must be… …product.
Example 2.1.8. Let {En }∞
n=1 be a collection of Hilbert A-modules for a C…

We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do…

► Any manned space mission must provide breathable air to its crew. For this reason, air leaks in spacecraft pose a danger to the mission…
(more)

▼ Any manned space mission must provide breathable air to its crew. For this reason, air leaks in spacecraft pose a danger to the mission and any astronauts on board. The purpose of this work is twofold: the first is to address the issue of air pressure loss from leaks in spacecraft. Air leaks present a danger to spacecraft crew, and so a method of finding air leaks when they occur is needed. Most leak detection systems localize the leak in some way. Instead, we address the identification of air leaks in a pressurized space module, we aim to determine the material in which the leak occurs. This is done with methods centered on statistics and machine learning.
In addition to these findings, we investigate one of the methods used in the leak identification section, the Hilbert-Huang Transform. This method has seen many demonstrations of its effectiveness, however this method lacks a solid theoretical framework. We make some contributions to the background of the Hilbert-Huang Transform.
Advisors/Committee Members: Ali Abedi, Andrew Knightly, Nigel Pitt.

…18
2.5
The process of Hilbert Spectral Analysis. . . . . . . . . . . . . . . . . .
20… …2.6
The Hilbert Transform, mapping a real valued function into the
complex plane… …is slow and does nothing to determine the location of the air leak. On
the ISS, modules may… …developing methods for leak identication and localization that I rst
came across the Hilbert-Huang… …very chaotic and we
sought a better method. This lead to the Hilbert-Huang Transform, which…